As Skyegirl1999 noted, you will need it for F! FP (this isn't in MaxPass). Otherwise, no, you don't "really" need it. You can make a screenshot of your ticket for your MaxPass redemption. That being said, I usually just use my paper pass for MP redemption, as it seems to be fastest for my group.
Hang on to your paper ticket just in case something goes wrong with your phone. Then you will need that paper ticket to get non-digital FPs, to scan at the FP return scanners, and (maybe) to rescan into your app in case your app goes wonky (this has happened to a few DISers). If you want to save your phone battery, you can use your paper ticket to scan at the FP return lines. Some people find that this helps, others find it doesn't make a difference.
Keep in mind that wi-fi is not great in the parks. It really is just a collection of hot spots, so reception can be spotty. If you're using only your phone, taking a screenshot of your FPs is helpful so you don't end up struggling with reception at the ride's entrance.
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